Theorem fusgrvtxfi 29299

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29298	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 496	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6499  Fincfn 8895  Vtxcvtx 28976  USGraphcusgr 29129  FinUSGraphcfusgr 29296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-fusgr 29297

This theorem is referenced by:  fusgrfupgrfs  29311  nbfusgrlevtxm1  29357  nbfusgrlevtxm2  29358  nbusgrvtxm1  29359  uvtxnm1nbgr  29384  cusgrm1rusgr  29563  wlksnfi  29887  fusgrhashclwwlkn  30058  clwwlkndivn  30059  fusgreghash2wsp  30317  numclwwlk3lem2  30363  numclwwlk4  30365